Flatness-Based Torque Control of Saturated Surface-Mounted ......Blondel-Park transformation...

This document contains a post-print version of the paper

Flatness-Based Torque Control of Saturated Surface-Mounted PermanentMagnet Synchronous Machines

authored by D. Faustner, W. Kemmetmüller, and A. Kugi

and published in IEEE Transactions on Control Systems Technology.

The content of this post-print version is identical to the published paper but without the publisher’s final layout orcopy editing. Please, scroll down for the article.

Cite this article as:D. Faustner, W. Kemmetmüller, and A. Kugi, “Flatness-based torque control of saturated surface-mounted permanentmagnet synchronous machines”, IEEE Transactions on Control Systems Technology, vol. 24, no. 4, pp. 1201–1213, 2016,issn: 1063-6536. doi: 10.1109/TCST.2015.2501345

BibTex entry:@ArticleFaustner16,Title = Flatness-Based Torque Control of Saturated Surface-Mounted Permanent Magnet Synchronous Machines

,Author = Faustner, D. and Kemmetm\"uller, W. and Kugi, A.,Journal = IEEE Transactions on Control Systems Technology,Pages = 1201--1213,Volume = 24,Year = 2016,Number = 4,Doi = 10.1109/TCST.2015.2501345,ISSN = 1063-6536

Link to original paper:http://dx.doi.org/10.1109/TCST.2015.2501345

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http://dx.doi.org/10.1109/TCST.2015.2501345


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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. ??, NO. ?, DECEMBER 201? 1

Flatness-based Torque Control of SaturatedSurface-Mounted Permanent Magnet Synchronous

MachinesDavid Faustner, Student Member, IEEE, Wolfgang Kemmetmuller, Member, IEEE,

and Andreas Kugi, Member, IEEE

Abstract—Modern permanent magnet synchronous machines(PMSMs) may also be operated in regimes where significantmagnetic saturation occurs. Classical fundamental wave modelsdo not incorporate magnetic saturation in a systematic way.Mostly only heuristic extensions can be found in literature.Control schemes based on such dq0-models are thus often unableto achieve the given control demands. This paper proposes aflatness-based torque control strategy for a saturated surface-mounted permanent magnet synchronous machine. Different toexisting works, magnetic saturation is considered in a thoroughand physically consistent way. Based on a calibrated magneticequivalent circuit model of the PMSM, a simplified model suitablefor the flatness-based controller design is derived. The proposedtwo-degrees-of-freedom control scheme inherently accounts forthe mutual coupling of the phase windings. Furthermore, thetime-varying controller gains are obtained by pole placementtechnique. In contrast to the majority of controllers used forPMSM control, the proposed control scheme is formulatedin the stator-fixed reference frame and hence no coordinatetransformation is necessary. The flatness-based torque controlleris implemented on an experimental test-bench. An acceleratedrun-up of the PMSM with about four times the rated torque isperformed to highlight the feasibility of the proposed approach.Finally, the flatness-based torque controller is compared with acommon vector control implementation based on a dq0-model ofthe motor.

Index Terms—Electric motor, permanent magnet motor, mag-netic saturation, magnetic equivalent circuit, flatness-based con-trol, torque control, optimization, experimental validation.

I. INTRODUCTION

ELECTRIC drive systems composed of permanent mag-net synchronous machines (PMSMs) are often used in

various technical applications, e.g., industrial machine tools,traction drives or automotive applications. This is mainly dueto their high performance and high energy efficiency. Model-based controller designs for PMSMs are primarily based onfundamental wave models. The main assumptions in thesemodels are sinusoidally distributed flux linkages of the statorwindings, a magnetic linear behavior of the iron core material(i.e. no saturation effects) and negligible iron losses (includingboth eddy current and hysteresis losses). Application of theBlondel-Park transformation directly yields the well-knowndq0-model. In this model, the electromagnetic quantities areindependent of the rotor angle and hence lower bandwidths are

The authors are with the Automation and Control Insti-tute, Technische Universitat Wien, Vienna, Austria (email:faustner,kemmetmueller,[email protected]).

Manuscript received June 8, 2015; accepted October 17, 2015.

sufficient for the current controllers. Several control strategieshave been developed based on classical dq0-models, wherethe most common is known as field-oriented control (FOC),see, e.g., [1] and [2]. Field-oriented control, also referred toas vector control, typically combines a dynamic decouplingscheme with two proportional-integral (PI) controllers to con-trol the motor currents along desired reference trajectories.More advanced control strategies comprise feedback lineariza-tion control [3]–[6], backstepping methods [7]–[9], passivity-based control [10]–[12], sliding-mode approaches [13]–[15]and model-predictive control [16]–[18]. Another frequentlyused method, which is not based on the dq0-transformation,is known as direct-torque control (DTC), see, e.g., [19]–[21].

Although control systems based on fundamental wave mod-els have been successfully implemented in a variety of appli-cations, there seems to be a paradigm shift in recent years.Traditionally, motors were designed in such a way that the useof fundamental wave models is justified by the construction. Inrecent years, the emphasis has been set on strategies to reducethe overall costs, for example by motor designs which aremore suitable to machine production. Amongst others, suchmeasures comprise the replacement of integer-slot windingsby fractional-slot concentrated windings and the applicationof simpler rotor structures, e.g., rotors with internal-mountedmagnets (IPMSM). Besides manufacturing aspects, PMSMsare often designed to generate a maximum torque for athermal-limited short-time operation in the overload range.Therein, almost all PMSMs suffer from significant magneticsaturation. The aforementioned aspects can result in inhomo-geneous air gap geometries, non-sinusoidal flux linkages andiron saturation. As a consequence, the assumptions of the fun-damental wave model are more or less violated for many actualmotor designs. Therefore, suitable mathematical models thataccurately describe these motors are required. Furthermore,advanced (model-based) control strategies are strived for toguarantee high control performance in all operating points ofthe machine.

In literature, to the best of the authors’ knowledge, nosystematic model-based approach for the design of appropriatecontrol strategies has been presented that inherently incorpo-rates spatial distributions of the flux linkages and magneticsaturation in a thorough and physically consistent way. Almostall presented works concerning the non-ideal behavior ofmodern PMSMs rely, more or less, on heuristic extensionsof the fundamental wave model. Several works have been

Post-print version of the article: D. Faustner, W. Kemmetmüller, and A. Kugi, “Flatness-based torque control of saturated surface-mountedpermanent magnet synchronous machines”, IEEE Transactions on Control Systems Technology, vol. 24, no. 4, pp. 1201–1213, 2016, issn:1063-6536. doi: 10.1109/TCST.2015.2501345The content of this post-print version is identical to the published paper but without the publisher’s final layout or copy editing.



published on how to reduce torque and speed ripples causedby the non-sinusoidal flux distributions, in particular at lowrotor speeds, see, e.g., [22]–[27]. Most of these works arebased on harmonic extensions of the fundamental wave ofthe flux linkages, either calibrated with offline measurements,finite-element (FE) results or online estimation strategies.Magnetic saturation, however, is neglected in these works. Acommon approach to include magnetic saturation is to varythe electromagnetic quantities (i.e. the inductances and thepermanent magnet fluxes) of the fundamental wave model withrespect to the load (i.e. currents), see, e.g., [28]–[33]. Again,these models are calibrated numerically or experimentally. Aconsequence of this approach is that the exact physical mean-ing of the derived model quantities is to a certain extent lost.To be consistent with physical principles, advanced controlstrategies based on comprehensive mathematical models seemto be promising to exploit the overall motor performance inthe whole operating range.

This paper presents a flatness-based torque control strat-egy for a surface-mounted permanent magnet synchronousmachine (SMPMSM). The operating range includes overloadoperation where significant magnetic saturation is present.Starting with a calibrated magnetic equivalent circuit model(MEC) in Section II, a simplified model is derived in Sec-tion III in view of the demands of the controller design.The proposed two-degrees-of-freedom torque control strategydeveloped in Section IV is composed of a flatness-basedfeedforward controller and a time-variant feedback error con-troller. The controller is implemented on a test bench usingan industrial electric drive system and is compared with acommon vector control implementation. The correspondingresults are presented in Section V.

II. MODELING OF PMSM

The PMSM under consideration is equipped with a rotorcomprising 10 surface-mounted neodymium-iron-boron (Nd-FeB) magnets (number of pole pairs p = 5), which arealternately magnetized. Fig. 1 shows a cross-sectional viewof the PMSM. The stator teeth are equipped with individualstator coils (number of windings Nc = 70), where threeconsecutive coils are connected in series to form single phasewindings (labeled a, b and c). The three phase windings arewye-connected and the neutral point is inaccessible (isolatedneutral). Such winding diagrams are usually called fractional-slot concentrated windings.

Besides numerically expensive methods like FE, the system-atic incorporation of complex geometries and magnetic satura-tion into electric machine models still represents a challengingtask. In this context, MEC modeling serves as a promisingand powerful tool to account for these challenges with amanageable modeling complexity. Although MEC modelingis frequently used in the design of electric machines, it hasnot been exploited for the controller design. The basic idea ofMEC modeling is the representation of the electromagneticbehavior in form of a network composed of concentratedelements, namely magnetic conductances (permeances) andmagnetomotive force (mmf) sources. The permeances cover

phase aphase c

phase b

statorrotor

magnets

Fig. 1. Cross-sectional view of the PMSM under consideration.

the specific geometry and material behavior, whereas the mmfsources represent the stator coils and rotor magnets.

In [34], a framework for MEC modeling of a PMSM withinternal-mounted magnets was proposed that allows for thecalculation of the model equations by means of graph theory,with a systematic choice of a minimal set of state variables andthe systematic incorporation of the electrical interconnectionof the stator coils. Because the suggested framework is notlimited to a specific motor design, this procedure was alsoapplied to the surface-mounted PMSM considered in thispaper. Subsequently, the main results are briefly summarizedto introduce the set of equations used. For details, the readeris referred to [35].

The permeance network proposed in [35] consists of mmfsources of the stator coils and rotor magnets, magneticallynonlinear iron permeances of the stator teeth and yoke,magnetically linear leakage permeances in the slot-openingareas between adjacent teeth as well as rotor angle dependentmagnetically linear air gap permeances covering the magneticcoupling between the stator and the rotor. The systematicderivation of the network is based on graph theory, wherea suitable tree has to be defined, which covers all nodes ofthe network without forming meshes. Additionally, all mmfsources have to be included in the spanning tree, cf. [35].Elements outside the tree form the co-tree. The fluxes of thetree φt are grouped into coil fluxes φtc ∈ R9×1, fluxes ofthe permanent magnets φtm ∈ R18×1 and permeance fluxesφtg ∈ R18×1. The co-tree only contains permeance fluxesφc ∈ R27×1. The same partitioning is performed for the mmfs,i.e. utc ∈ R9×1, utm ∈ R18×1, utg ∈ R18×1 and uc ∈ R27×1.The mmfs of the coils are expressed as utc = NcVc itc,with the winding matrix Nc = diag(Nc) and the independentelectric phase currents itc = [ia, ib]

T , fulfilling ia+ib+ic = 0due to the isolated neutral point. The non-square intercon-nection matrix Vc represents the electrical interconnection ofthe individual stator coils and also determines the vector ofindependent electric voltages vtc = [vac, vbc]

T in the form

iTtcvtc =(itc)T

(Vc)Tvtc =

(itc)T

vtc, (1)




with the independent electric line-to-line voltages vac = va −vc and vbc = vb − vc. The (constant) mmfs of the magnetsare determined by the coercive field strength and the magnetheight. The magnetic interconnection of the elements of thepermeance network can be systematically described with theincidence matrix D ∈ R45×27, which, for convenience, isfactorized in the form DT = [DT

c ,DTm,D

Tg ] and composed

of elements −1, 1, 0. Please note that the incidence matrixD defines the fundamental relationship

[φtuc

]=

[D 00 −DT

] [φcut

](2)

of the network, with the magnetic fluxes φTt = [φTtc,φTtm,φ

Ttg]

and the mmfs uTt = [uTtc,uTtm,u

Ttg] of the tree elements

and the magnetic fluxes φc and the mmfs uc of the co-treeelements.

The constitutive equations, i.e. the relationships between themmfs and the fluxes of the permeances of the tree and co-tree,are given by

[φtgφc

]=

[Gt 00 Gc

] [utguc

], (3)

where the elements of the diagonal permeance matrices of thetree Gt ∈ R18×18 and co-tree Gc ∈ R27×27 depend eitheron uc or ut due to saturation (iron permeances), on the rotorangle ϕ (radial airgap permeances) or are constant (leakagepermeances).

Substitution of (3) into (2), consideration of the introducedpartitioning of the mmfs and fluxes of the tree and co-treeas well as a straightforward rearrangement directly results inthe machine model with current input of the PMSM, whichdescribes the influence of the electric currents and the rotorangle ϕ on the mmfs and magnetic fluxes in form of a set ofnonlinear algebraic equations

I 0 DcGcD

Tg

0 I DmGcDTg

0 0 Gt + DgGcDTg

φtcφtmutg

=

−DGc

(DTc utc + DT

mutm).

(4)

Therein, the unknown variables are the coil fluxes φtc, thefluxes of the permanent magnets φtm and the mmfs utg ofthe permeances of the tree. The inputs on the right-hand sideof (4) are the mmfs of the coils utc and the (constant) mmfsof the magnets utm. If the electric currents itc are known fora given rotor angle ϕ, (4) can be numerically solved for utgand the fluxes can be simply calculated from linear equations.

Based on co-energy considerations of the elements of thepermeance network and due to the fact that the chosen networkonly exhibits a dependence on the rotor angle in the radial airgap permeances, the developed electromagnetic torque can beexpressed as, see [34],

τ =1

2uTtg

∂Gt

∂ϕutg +

1

2uTt D

∂Gc

∂ϕDTut. (5)

Modern electric drive systems are typically equipped with avoltage source inverter (VSI) triggered by pulse-width mod-ulation (PWM). Thus, the electric voltages at the machineterminals serve as control inputs and the flux dynamics have

to be added to obtain a machine model with voltage input.In [34] and [35], a comprehensive analysis and a systematicelimination of magnetic and electric redundancies was carriedout to obtain a set of differential-algebraic equations (DAE)of minimum dimension. Subsequently, a slightly different andmore descriptive approach is introduced to obtain a reduced setof differential equations for the corresponding flux linkages.

Consider a wye-connected three-phase winding system.Application of Faraday’s induction law directly yields

d

dtψabc = −Riabc + (vabc − vn) , (6)

with the flux linkages ψabc = [ψa, ψb, ψc]T of the phase

windings, the phase currents iabc = [ia, ib, ic]T , the phase

winding resistances R = diag (R), and the terminal and neu-tral point voltages vabc = [va, vb, vc]

T and vn = [vn, vn, vn]T

with respect to the reference potential of the VSI. Due to theisolated neutral point, only two phase currents are independent.Hence, there is a redundancy in the voltage equation (6). Thisredundancy can be systematically eliminated by the invertibletransformation matrix

Vv =

1 0 −10 1 −11 1 1

. (7)

Left multiplying the voltage equation (6) with Vv yields

d

dt

[ψa − ψcψb − ψc

]

︸︷︷︸ψItc

= −[2R RR 2R

]

︸︷︷︸R

[iaib

]

︸︷︷︸itc

+

[va − vcvb − vc

]

︸︷︷︸vtc

(8a)

d

dt(ψa + ψb + ψc) = −R(ia + ib + ic︸︷︷︸

=0

) + (v0 − 3vn), (8b)

with v0 = va + vb + vc. In (8b), the sum of the flux linkagesvanishes for an ideal symmetric three-phase machine withoutleakage fluxes. In this case, the neutral point voltage vn isdetermined by forcing the voltage term in (8b) to zero (i.e.floating neutral). In a real machine, however, ψa + ψb + ψcdoes not necessarily have to be zero. In any case, the set ofdifferential equations (8a) and (8b) are decoupled and neitherψa + ψb + ψc nor v0 and vn have an influence on the torque.Thus, (8b) is of no interest for the further derivations. Pleasenote that an equivalent formulation of (8a) can be obtained bya more general analysis of Vc and D as given in [34] andapplied in [35]. The vector of independent flux linkages ψ

I

tc

and the resistance matrix R in (8a) can be calculated in theform

ψI

tc = −VTc Ncφtc (9a)

R = 1/3VTc RVc. (9b)

In conclusion, the machine model with voltage input ofthe PMSM is given by the DAE system (8a), (4) and (5).If the DAE system (8a) and (4) is intended to be used fordynamical simulations with the flux linkages as state variablesand the electric voltages as inputs, the independent electricphase currents itc = [ia, ib]

T have to be calculated from theset of nonlinear algebraic equations (4) for given flux linkages.In order to do so, the magnetic and electric redundancies




still present in (4) have to be eliminated. A comprehensivediscussion on how this can be done systematically is given in[34] and applied to the considered PMSM in [35]. However,as it will be shown in the subsequent sections, this reduction isnot necessary for the design of a controller, if the independentelectric phase currents can be measured.

To demonstrate the quality of the model and thus the feasi-bility of the modeling approach, some simulation and experi-mental results are briefly recapitulated, see [35]. Thereby, thenominal model was parameterized by geometric and materialdata and calibrated by measurements. To validate the machinemodel with current input (4) and (5), the PMSM was suppliedwith direct current ia = −ib, ic = 0 A and driven by aharmonic drive system at a constant rotational speed. Thespeed was chosen sufficiently low, such that the influence ofthe back electromotive force (back-emf) is negligible. Therotor angle ϕ was measured by a high resolution encoderand the torque τ was measured using a torque transducer.Fig. 2(a) shows the torque as a function of the rotor anglefor ia = −ib = 5 A, ic = 0 A. A very good agreementof both the shape and the amplitude of the simulated andmeasured torque can be identified. As can be further seenin Fig. 2, the main electromagnetic quantities including thecurrents, the voltages, the flux linkages and the torque show aperiodicity of 72, which results from the number of pole pairs(p = 5). Fig. 2(b) depicts the results for ia = −ib = 20 A,ic = 0 A. This increased current leads to a significant influenceof magnetic saturation of the iron core, which can be seenas the corresponding torque does not increase linearly withthe current (as would be the case for an unsaturated motor).The proposed model still accurately resembles the measuredtorque. To evaluate the influence of magnetic saturation onthe amplitude of the torque, this experiment was repeatedfor different operating points. The resulting current-to-torquecharacteristic is depicted in Fig. 3. The strong influence ofmagnetic saturation can be recognized and it can be seenthat it is accurately reflected by the proposed mathematicalmodel (4) and (5). To validate the dynamical behavior ofthe developed model, i.e. the effect of the electric voltageson the electric currents, the induction law (8a) is considered.In the first experiment, the PMSM was driven at rated speedn = 3000 rpm by an external speed-controlled machine andthe back-emf was measured at open machine terminals (i.e.ia = ib = ic = 0 A). The results of this open-circuitexperiment in Fig. 2(c) again show a high accuracy of theproposed machine model. In the final experiment, a short-circuited motor, i.e. va− vc = 0, vb− vc = 0 was considered,where the PMSM was driven at a constant speed of n = 1600rpm. The measurement results of the currents ia, ib and ic arecompared with the simulation results in Fig. 2(d). The verygood agreement in this experiment also proves the feasibilityof the chosen modeling approach. Looking at the results ofFig. 2, it might seem that the quantities are purely sinusoidal.A fast Fourier transform (FFT) analysis, however, exhibits thepresence of harmonics of order k = 5, 7, 11, 13, . . . with rapiddescending amplitudes.

The proposed machine model with voltage input of thePMSM in form of the DAE system (8a), (4) and (5) accurately

−8

−4

0

4

8

torq

ueτ

inNm

(a)

measurementsimulation

−24

−12

0

12

24

torq

ueτ

inNm

(b)

−450

−300

−150

0

150

300

450

vbcvca vab

volta

gein

V

(c)

0 12 24 36 48 60 72−12

−8

−4

0

4

8

12

iaic ib

rotor angle ϕ in

curr

ent

inA

(d)

Fig. 2. Comparison of the developed model with measurements: torque asa function of the rotor angle ϕ in (a) for ia = −ib = 5 A and in (b) foria = −ib = 20 A, induced voltages due to open-circuit operation in (c)for n = 3000 rpm and the currents due to short-circuit operation in (d) forn = 1600 rpm.




0 6 12 18 24 300

6

12

18

24

current ia in A

torq

ueτ

inNm

measurementsimulation

Fig. 3. Comparison of the measured and simulated current-to-torque charac-teristic for a current supply in the form ia = −ib, ic = 0 A.

reflects the motor behavior in the entire operating range. Inparticular, the model inherently accounts for a non-sinusoidalflux distribution and magnetic saturation in overload operation.Based on this comprehensive machine model, a simplifiedmodel of the PMSM suitable for flatness-based control designwill be derived in the next section.

III. OPTIMAL CURRENTS AND FLUX LINKAGES

In this paper, a flatness-based torque control strategy ofthe PMSM in the entire operating range including overloadoperation with significant magnetic saturation is discussed. Aclassical torque control loop utilizing a torque transducer isundesirable in most applications due to additional hardwareand cost. Thus, indirect torque control by proper currentcontrol will be considered as an appropriate solution. In a firststep, optimal currents as functions of the rotor angle ϕ for adesired torque are calculated by minimizing the Joule lossesof the motor. Using the set of nonlinear algebraic equationsfor utg from (4), i.e.(Gt + DgGcD

Tg

)utg = −DgGc

(DTc utc + DT

mutm)

(10)

with utc = NcVc itc, the nonlinear constrained optimizationproblem reads as

minitc,utg

(itc)T (itc)

s.t. eq. (10)

τd (ϕ)− τ (ϕ) = 0,

(11)

with the desired torque τd(ϕ), the calculated torque τ(ϕ) from(5) and itc = [ia, ib]

T . Minimization of the Joule losses is alsocommonly known as maximum torque per ampere (MTPA).An active-set algorithm is used to calculate a numericalsolution of the optimal currents as functions of the rotor angleϕ from the nonlinear constrained optimization problem (11).For this, the rotor angle ϕ is discretized by 144 points whichcorresponds to a step size ∆ϕ = 0.5 mechanical angle.Corresponding results are shown in Fig. 4, where in Fig. 4(a)the desired torque τd = 5 N m was chosen close to the ratedvalue of the motor. The associated optimal current shape is,in fact, almost perfectly sinusoidal. If the desired torque is

−4.5

−3

−1.5

0

1.5

3

4.5

icib ia

curr

ent

inA

(a)

optimalfundamental

0 12 24 36 48 60 72−32

−24

−16

−8

0

8

16

24

32

icib ia

rotor angle ϕ in

curr

ent

inA

(b)

Fig. 4. Optimal current shape and corresponding fundamental wave providingminimum Joule losses for a constant desired torque of τd = 5 N m in (a)and τd = 25 N m in (b).

increased towards overload operation (i.e. τd = 25 N m) asshown in Fig. 4(b), harmonics of order k = 5, 7, 11, 13, . . .are more developed. The fundamental wave, however, is stilldominating. This behavior is quite common for PMSMs withsurface-mounted magnets, where magnetic saturation stronglyinfluences the amplitude of the currents, while the shape stillis close to a sinusoidal form. Other motor designs as PMSMswith internal-mounted magnets frequently show dominatinginfluence of harmonics already at low torques, which makesthe subsequent control strategy not directly applicable to thesemotor designs. The subsequent analysis and controller designare based on a fundamental wave approximation of the optimalcurrent shapes, which, of course, is at the expense of atorque error. The small and thus tolerable influence of thisapproximation will be investigated later in the paper.

It should be noted that it is also possible to considersinusoidal currents right from the start and to optimize theamplitude and phase of the current by minimizing the resulting(cumulative) torque error over the discretized fundamentalperiod of the rotor angle in the cost function. This approachhas not been chosen due to the following reasons: (i) Ingeneral, sinusoidal currents do not yield an almost constanttorque, e.g., IPMSM with distinctive nonuniform air gap ge-ometry. (ii) The advantage of the chosen approach (to calculateoptimal current waveforms first, followed by a sinusoidalapproximation) over the above-mentioned approach appears




0 5 10 15 20 25 30 35 400

5

10

15

20

25

current amplitude id in A

torq

uein

Nm

Fig. 5. Current-to-torque characteristic due to a sinusoidal current supplyaccording to (12).

in a significant reduction of the dimension of the nonlinearconstrained optimization problem in terms of optimizationvariables and equality constraints with almost the same ac-curacy.

The chosen optimization and approximation leads to thecurrent-to-torque characteristic as given in Fig. 5. Here again,the nonlinear characteristic due to magnetic saturation athigher loads can be recognized. The fundamental wave ap-proximation of the optimal currents can be expressed as

idtc = [ida, idb ]T = idTc,i(ϕ), (12)

with Tc,i(ϕ) = [cos(α), cos(α+2π/3)]T and α = p(ϕ−ϕ1),the amplitude of the current id(τd) according to the current-to-torque characteristic in Fig. 5 and the phase ϕ1 = 40 +18sign(τd). If the currents in (12) are supplied to the ma-chine, it can be expected that the PMSM generates the desiredtorque τd. Since the electric voltages at the machine terminalsserve as control inputs, the voltage equation (8a) has to beconsidered to account for the corresponding flux dynamics.

Substituting the optimal sinusoidal currents (12) into (4)allows for the numerical calculation of the corresponding coilfluxes φtc and, based on the relationship (9), the flux linkagesψac = ψa − ψc and ψbc = ψb − ψc. This procedure appliedfor different amplitudes id and rotor angles ϕ directly yieldstwo-dimensional maps for the flux linkages ψac(i

d, ϕ) andψbc(i

d, ϕ), see Fig. 6 for a map of ψac. For the flux linkagecalculation, a grid with ∆ϕ = 0.5 (a total of 144 grid points)and ∆i = 2 A (a total of 20 grid points) is chosen. The fluxlinkages ψac and ψbc have identical form but their phase isshifted by ϕ, i.e. ψbc(id, ϕ) = ψac(i

d, ϕ+ ϕ) with ϕ = 12.To analyze the form of the flux linkages, profiles for

constant current amplitudes are depicted in Fig. 7. Besidethe rising amplitude of the flux linkages with increasing cur-rent, an additional load-dependent phase shift occurs. Again,harmonics of order k = 5, 7, 11, 13, . . . are visible in theoverload operating range. The second simplification step in theconsidered analysis consists of the sinusoidal approximationof the flux linkages in the entire operating range. Thus, thefundamental wave of the flux linkages due to the sinusoidal

0 18 36 54 010203040−0.4

−0.2

0

0.2

0.4

rotor angle ϕ in current in

A

fluxψac

inVs

Fig. 6. Two-dimensional map of the flux linkage ψac (id, ϕ) due to asinusoidal current supply according to (12).

0 12 24 36 48 60 72−0.6

−0.4

−0.2

0

0.2

0.4

0.6

id = 40A

id = 5A

rotor angle ϕ in

flux

inVs

optimalfundamental

Fig. 7. Profiles of the two-dimensional map of the flux linkage ψac (id, ϕ)for id = 5 A and id = 40 A.

current supply according to (12) can be formulated as

ψdac(id, ϕ) = ψd sin(p(ϕ−∆ϕ)) (13a)

ψdbc(id, ϕ) = ψdac(i

d, ϕ+ ϕ), (13b)

with a current-dependent amplitude ψd(id) and phase shift∆ϕ(id). These functions, as depicted in Fig. 8, are approxi-mated by means of polynomials of order three and five for thephase and the amplitude of the flux linkages, respectively.

Utilizing these approximations of the flux linkages in thevoltage equation (8a) results in

∂ψdac∂id

did

dt+∂ψdac∂ϕ

ω = −R(2ida + idb

)+ vac (14a)

∂ψdbc∂id

did

dt+∂ψdbc∂ϕ

ω = −R(ida + 2idb

)+ vbc, (14b)

with the electric line-to-line voltages vac = va − vc, vbc =vb − vc, the phase currents ida, idb in the form (12), and theangular velocity ω.

Up to now, optimal currents for a desired torque have beencalculated by a nonlinear constrained optimization problem.Based on a sinusoidal approximation of the optimal currents in




0 5 10 15 20 25 30 35 400

4

8

12

16


phas

esh

ift

∆ϕ

in

phaseamplitude

0.2

0.3

0.4

0.5

0.6

amplitude

ψd

inV

s

Fig. 8. Load-dependent phase shift and amplitude of the sinusoidal fluxlinkages (13).

(12), the corresponding flux linkages have been calculated anda sinusoidal approximation of these flux linkages yields thesimplified model (14). Note that the simplified model (14) stillincorporates the magnetic saturation effect in a systematic andphysically consistent way. Thus, this model can be regardedas a good basis for the flatness-based controller design in thenext section.

IV. CONTROLLER DESIGN

The proposed flatness-based torque control strategy relieson the measurements of the (independent) electric currentsitc = [ia, ib]

T and of the rotor angle ϕ. It is based on atwo-degrees-of-freedom control structure with a flatness-basedfeedforward controller in combination with a time-variantfeedback error controller. As pointed out in the previoussection, torque control is replaced by controlling the currentsalong the corresponding optimal trajectories. The control in-puts vac and vbc are split into

vac = vdac + vcac (15a)

vbc = vdbc + vcbc, (15b)

with the feedforward control inputs vdac, vdbc and the feedback

control inputs vcac, vcbc, respectively. Furthermore, the indepen-

dent electric phase currents itc = [ia, ib]T are written in the

form

ia = ida + ∆ia (16a)

ib = idb + ∆ib, (16b)

with the desired currents idtc = [ida, idb ]T due to (12) and the

tracking errors of the currents ∆ia and ∆ib. The flatness-basedfeedforward controller can be directly deduced from (14) inthe form

vdac = R(2ida + idb) +∂ψdac∂id

did

dt+∂ψdac∂ϕ

ω (17a)

vdbc = R(ida + 2idb) +∂ψdbc∂id

did

dt+∂ψdbc∂ϕ

ω, (17b)

with the sinusoidal flux linkages ψdac and ψdbc from (13), thedesired sinusoidal currents idtc from (12) in combination with

the desired current amplitude id and the nominal windingresistance R. The desired value of id and its time derivativeare calculated from the current-to-torque characteristic givenin Fig. 5, assuming at least a C1-continuous desired torquesignal τd(t).

The flatness-based feedforward controller (17) is extendedby a (time-variant) feedback error controller to stabilize thetracking error in case of parameter variations due to un-modeled dynamics, measurement uncertainties and externaldisturbances. In this work, unmodeled dynamics mainly resultsfrom iron losses (including both eddy current and hysteresislosses) as well as the temperature behavior of the magnets andwinding resistances. Measurement inaccuracies typically occurdue to the approximate numerical differentiation of the rotorangle ϕ when calculating the angular velocity ω, and the cur-rent measurement signals are often corrupted by measurementnoise and disturbances resulting from the switching of the VSI.Of course, the feedback controller also has to compensatefor variations resulting from the simplification steps in thesimplified model.

The derivation of the current error dynamics is based ona Taylor series expansion of (8a) and (4). In this context itis advantageous to express the flux linkages as functions ofthe independent electric phase currents itc = [ia, ib]

T . Notethat the desired current amplitude id in (12) can be easilycalculated from the desired phase currents ida and idb withthe help of ida + idb + idc = 0. Using (16), the flux linkagesψac(ia, ib, ϕ) and ψbc(ia, ib, ϕ) can be written in the formψac = ψdac + ∆ψac, ψbc = ψdbc + ∆ψbc, with[∆ψac∆ψbc

]≈[∂ψac(i

da,i

db ,ϕ)

∂ia

∂ψac(ida,i

db ,ϕ)

∂ib∂ψbc(i

da,i

db ,ϕ)

∂ia

∂ψbc(ida,i

db ,ϕ)

∂ib

]

︸︷︷︸Lψ

[∆ia∆ib

]. (18)

Given a sufficiently accurate current control, deviations fromthe desired operating point idtc are small and thus the errormade by this approximation is also small. Substitution of (15)–(18) into the voltage equation (8a) results in

d

dt

[∆ψac∆ψbc

]=

d

dtLψ

[∆ia∆ib

]+ Lψ

d

dt

[∆ia∆ib

]

= −[2R RR 2R

] [∆ia∆ib

]+

[vcacvcbc

].

(19)

As can be seen from (18), the differential inductance matrixLψ depends on the operating point. For a more detailedanalysis, the partial derivatives of the flux linkages ψac and ψbcwith respect to the currents ia and ib have to be evaluated at thedesired sinusoidal reference trajectories ida and idb according to(12). Since explicit analytical expressions of these terms arenot available, appropriate approximations have to be derived.

It can be shown that the derivatives of the permeancematrices Gt and Gc with respect to itc and utg are small andthus can be neglected. Please note that in the magneticallylinear case, these derivatives would be exactly zero. For theconsidered magnetically nonlinear case, the derivatives aresmall, since the rate of change of the relative permeabilityµr(H) of the iron core with respect to the magnetic fieldstrength H is also small. Thus, the errors of this approximation




0 36 72 108 144 180 216

10

15

20

25

30

35

72 144

τd = 5 N m τd = 15 N m τd = 25 N m

rotor angle ϕ in

indu

ctan

cein

mH

(a)

∂ψac∂ia∂ψac∂ib

0 5 10 15 20 25 30

10

15

20

25

30


indu

ctan

cein

mH

(b)

LsLm

Fig. 9. Numerical results of the differential inductance matrix Lψ for differentoperating points as a function of the rotor angle ϕ in (a) and average valuesover a fundamental period of 72 in (b).

are again small and thus this approximation can be consideredfeasible. Then, (4) yields the approximations

∂φtc∂ itc

≈ −DcGcDTc NcVc −DcGcD

Tg

∂utg∂ itc

(20a)

∂utg∂ itc

≈ −(Gt + DgGcDTg )−1DgGcD

Tc NcVc. (20b)

With (20) and (9), the differential inductance matrix Lψ readsas

Lψ = −VTc Nc

∂φtc∂ itc

. (21)

A numerical solution of Lψ as a function of the rotor angleϕ and for fixed values of id can be easily determined from(21). Numerical results for different values of id (i.e. τd)are depicted in Fig. 9(a). The calculations show that thedifferential mutual inductances ∂ψac/∂ib and ∂ψbc/∂ia (off-diagonal entries of Lψ) are identical, which, as is expected,implies that Lψ is a symmetric matrix. Additionally, averagevalues over a fundamental period of 72 of the entries ofthe diagonal elements ∂ψac/∂ia and ∂ψbc/∂ib (differentialself-inductances), are identical too. It can be seen fromFig. 9(a) that the differential self and mutual inductancesshow a periodicity of twice the fundamental period, which

is physically interpretable. The variation of the differentialinductance matrix Lψ with respect to the rotor angle ϕ isneglected for the design of the feedback controller. One candeduce from Fig. 9(a) that this assumption is clearly feasiblefor desired torques up to 15 N m, but the variations of thedifferential inductances increase for higher torques. However,as will be shown subsequently, Lψ only acts as a gain in thefeedback control law and thus the decrease of control accuracymade by this assumption is again negligible. Hence, operatingpoint dependent average values over a fundamental period of72, as shown in Fig. 9(b), are used instead. Here, the stronginfluence of magnetic saturation of the iron core and a ratioLs/Lm ≈ 2 can be observed. Thus, the averaged differentialinductance matrix Lψ can be formulated as

Lψ =

[Ls

12 Ls

12 Ls Ls

]. (22)

Substitution of Lψ into (19) allows for further simplifications.As can be deduced from Fig. 9(b), the variation of Lψ withrespect to the current amplitude is rather small even if the cur-rent amplitude is changed in a wide range. Consequently, thetime derivative of Lψ is rather small. Furthermore, assuming aproper feedback controller, the current errors ∆ia and ∆ib aresmall as well. As a consequence, the product of these termscan be neglected in (19) without significant loss of accuracy.It should be noted that the approximations performed in thissection are also applicable to many other PMSMs with surface-mounted magnets. The current error dynamics (19) may thusbe written in the simplified form

Lψd

dt

[∆ia∆ib

]

︸︷︷︸∆itc

= −[2R RR 2R

]

︸︷︷︸R

[∆ia∆ib

]+

[vcacvcbc

]

︸︷︷︸vctc

, (23)

where the averaged differential inductance matrix Lψ is afunction of the desired current amplitude id and hence the errordynamics (23) is time-variant. Application of the feedbackcontrol law

vctc = R∆itc − Lψ

(kp∆itc + ki

∫∆itcdt

)(24)

directly yields the second-order closed-loop error dynamics

d2

dt2∆itc + kp

d

dt∆itc + ki∆itc = 0. (25)

The positive controller gains kp = diag(kp) and ki =diag(ki) can be calculated by choosing desired eigenvalues ofthe error dynamics (25). The overall control law of the PMSMis composed of the flatness-based feedforward controller (17)and the time-variant feedback error controller (24). A blockdiagram of the proposed control structure is depicted inFig. 10.

V. EXPERIMENTAL RESULTS

This section demonstrates the quality and feasibility ofthe proposed flatness-based torque controller. The controlleris compared with a common vector control implementationbased on the well-known dq0-model of the motor. Both




τd id

id

ddt i

d

ϕ, ωτd

vdtc

vctc

vtc

ϕ, ω

itc

τd

PMSMeq.(17)

eq.(24)

Fig. 5 Filter

Fig. 10. Block diagram of the proposed flatness-based torque control structure,comprising the feedforward and feedback part.

control strategies are implemented on the DSPACE realtime-platform DS1103 triggering an industrial standard VSI withinsulated-gate bipolar transistors (IGBTs). A sampling timeof Ta = 50 µs of the controller and a PWM frequency offp = 20 kHz were used in the experiments. The rotor angleϕ is measured with a motor-shaft mounted encoder and theelectric phase currents itc = [ia, ib]

T are measured using LEMcurrent transducers. Furthermore, the static inverter lossesof the VSI were identified and compensated in the voltagegeneration process.

A. Vector control of the PMSM

In a classical dq0-model, magnetic linearity (i.e. no mag-netic saturation), negligible iron losses (including both eddycurrent and hysteresis losses) and fundamental waves areassumed. Application of the Blondel-Park transformation [2]in the form vdq0 = K(ϕ)vabc and idq0 = K(ϕ)iabc with

K(ϕ)=2

3

cos(ρ) cos(ρ+ 2π3 ) cos(ρ− 2π

3 )− sin(ρ) − sin(ρ+ 2π

3 ) − sin(ρ− 2π3 )

1/2 1/2 1/2

(26)

and ρ = pϕ to a fundamental wave model of the PMSMdirectly results in the dq0-model expressed in rotor-fixedcoordinates

Ldqd

dtidq = −Rdqidq − pωLdqidq − pωΨPM + vdq (27a)

τdq =3

2p(

ΨTPM idq + iTdqLdqidq

), (27b)

with the voltage vector vdq = [vd, vq]T , the current vector

idq = [id, iq]T , the magnetic flux vector ΨPM = [ψPM , 0]T

with ΨPM = JΨPM , the angular velocity ω, the number ofpole pairs p, the electromagnetic torque τdq , the inductancematrix Ldq = diag([Ld, Lq]) with Ldq = JLdq and J in theform

J =

[0 −11 0

]. (28)

The dynamics of the zero sequence components is decoupledand not presented in this work. Obviously, all electromagneticquantities in (27) are independent of the rotor angle ϕ.

TABLE IMAIN TECHNICAL DATA OF THE PMSM

Term Value UnitRated speed 3000 rpmRated torque 6.16 N mRated current 3.78 Arms

Torque constant kt 1.63 Nm/Arms

Voltage constant ke 98.43 V/1000min−1

Phase winding resistance R 1.245 Ω

Homogenous air gap geometries in surface-mounted PMSMsentail Ld = Lq . Application of the control law

vdq = Rdqiddq + pωLdqidq + pωΨPM + Ldq

d

dtiddq

− kp,dq(idq − iddq)− ki,dq

∫(idq − iddq)dt

(29)

directly yields the second-order closed-loop error dynamics

Ldqd2

dt2∆idq + (Rdq +kp,dq)

d

dt∆idq +ki,dq∆idq = 0, (30)

with the tracking error ∆idq = idq − iddq and the referencetrajectories iddq = [idd, i

dq ]T . The positive controller gains

kp,dq = diag(kp,dq) and ki,dq = diag(ki,dq) can be calculatede.g. by pole placement. For a surface-mounted PMSM (i.e.Ld = Lq), the torque τdq only depends on the q-axis current iq ,see (27b). Hence, the reference trajectory idq has to be chosenaccording to the desired torque τd in the form idq = τd/kt,with the torque constant kt = 3/2pψPM . The reference valuefor the d-axis current id is set to idd = 0 A to minimize Joulelosses. It should be noted that the control law (29) stronglydepends on an accurate knowledge of the model parameters.Moreover, the exact compensation of the voltage terms onthe right-hand side of (27a) is not assured in a practicalimplementation due to parameter variations and disturbances.Finally, the dq0-model with constant parameters is inaccuratewhen the PMSM is operated in the overload range, wheremagnetic saturation occurs.

For the practical implementation, typically a simplifiedversion of (29) in the form

vdq = pωΨPM−kp,dq(idq−iddq)−ki,dq∫

(idq−iddq)dt (31)

is implemented. This control strategy results from neglectingthe feedforward terms Rdqi

ddq , Ldq

ddt i

ddq and the induced

voltage term pωLdqidq due to the mutual coupling. Note thatthe remaining model parameters of the control law (31), alsocommonly known as voltage constant ke = pψPM , can beeasily obtained from the technical data sheet of the motor. Themain technical data of the PMSM are shown in TABLE I.

Eq. (27b) does not account for the nonlinear current-to-torque behavior. To improve torque tracking accuracy, it iscommon industrial practice to use a (measured) current-to-torque characteristic as given in Fig. 3 in order to calculatethe desired current idq as a function of the desired torque τd.This is also done in the experiments of the next subsection,which yields an improved torque tracking accuracy.




PMSM torque sensor inertia

Fig. 11. Test bench setup of the PMSM.

B. Benchmark Experiment

The experimental validation of the proposed flatness-basedtorque control strategy and the comparison with the vectorcontrol implementation (31) using the nonlinear current-to-torque characteristic in Fig. 5 are performed on a test benchsetup depicted in Fig. 11. The motor shaft is attached to atorque sensor and a load inertia (Jload ≈ 0.03 kgm2) by meansof torsionally stiff couplings. For the proposed flatness-basedtorque controller, both poles of the second-order closed-looperror dynamics (25) are placed at Λ = [−1650,−1650]T (theresulting controller gains kp and ki are, however, time-varying)and the controller gains of the vector control strategy (31) arechosen as kp,dq = diag(52.33) and ki,dq = diag(0.0024).Appropriate integrator anti-windup strategies are implementedfor both control schemes. As a benchmark experiment, anaccelerated run-up of the PMSM is performed. To do so,the motor is accelerated with the torque profile depicted inFig. 12(a). According to Fig. 5, a desired torque of τd =25 N m highly saturates the PMSM. As can be deduced fromFig. 12(a), the proposed flatness-based torque control strategygenerates a sufficiently smooth torque even at high speeds.It should be noted that Fig. 12(a) illustrates the recalculatedtorque according to (5) on the basis of the measured currentsitc = [ia, ib]

T and the measured rotor angle ϕ. This makessense, because the developed model is validated and calibratedwith measurements. The torque measured by the torque sensorcontains higher frequency oscillations caused by the elasticityof the drive-train and errors in the alignment of the mechanicalsetup. Thus, these measurements are not very illustrative andhave therefore not been included in the present manuscript. Acomparison of the averaged values, however, shows very goodagreement with the recalculated torque.

Comparing the results of the proposed flatness-based torquecontrol concept (τ ) with the vector control strategy (τdq)shows significantly increased fluctuations of ∼ 5 N m forthe vector control concept in comparison to less than 1 N mof the proposed control concept, see Fig. 12(a). To analyzethe reasons for this behavior, the desired and the actualcurrents idq , iq and idd, id are depicted in Fig. 12(b) andFig. 12(c), respectively. It can be seen that both currents showlarge variations around their desired values in case of thevector control strategy. This is due to the fact that in theconsidered experiment large magnetic saturation of the motoris present which is not covered by the simple dq0-model. It,however, seems that the higher frequency fluctuations do not

−30

−20

−10

0

10

20

30

torq

uein

Nm

(a)

τdqτ

−50−40−30−20−10

01020304050

curr

ent

inA

(b)

iqidq

−10

−5

0

5

10

curr

ent

inA

(c)

ididd

0 0.25 0.5 0.75 1−1.5

−1

−0.5

0

0.5

1

1.5×103

time in s

rota

tiona

lsp

eed

inrp

m

(d)

ndqn

Fig. 12. Accelerated run-up of the PMSM and comparison of the proposedflatness-based torque control strategy with a common vector control imple-mentation: (a) recalculated torque profile, (b) q-axis current, (c) d-axis current,(d) rotational speed. For all subplots, the abscissa is the time measured in s.




−30

−20

−10

0

10

20

30

40(I) (II) (III)

curr

ent

inA

(a)

iaida

−3

−2

−1

0

1

2

curr

ent

inA

(b)

∆ia

0 72 144 216 288 3600

0.2

0.4

0.6

0.8

1

72 144

rotor angle ϕ in

duty

cycl

e

(c)

χaχbχc

Fig. 13. Sections of the accelerated run-up of the PMSM utilizing theproposed flatness-based torque control strategy for different rotational speeds,i.e. n ≈ 500 rpm in (I), n ≈ 1000 rpm in (II) and n ≈ 1500 rpm in (III).For all subplots, the abscissa is the rotor angle measured in .

have a large influence on the resulting rotational speed, seeFig. 12(d). In fact, the large inertia of the load suppressesthese fluctuations. For applications with smaller inertia and inview of increased thermal losses of the VSI, the behavior ofthe vector control strategy turns out to be rather problematicin industrial applications.

In order to compare both control strategies with respectto their ability to track desired reference trajectories of themotor currents, the d-axis and q-axis currents idq of the vectorcontrol strategy are transformed into phase currents itc,dq =[ia,dq, ib,dq]

T by the inverse of the transformation matrix (26).First, the results of the proposed control strategy as depicted inFig. 13 are discussed. In this figure, the quantities are plotted

−40

−30

−20

−100

10

20

30

40

50(I) (II) (III)

curr

ent

inA

(a)

ia,dqida,dq

0 72 144 216 288 3600

0.2

0.4

0.6

0.8

1

72 144

rotor angle ϕ in

duty

cycl

e

(b)

χa,dq

Fig. 14. Sections of the accelerated run-up of the PMSM utilizing a commonvector control implementation for different rotational speeds, i.e. n ≈ 500rpm in (I), n ≈ 1000 rpm in (II) and n ≈ 1500 rpm in (III). For bothsubplots, the abscissa is the rotor angle measured in .

as a function of the rotor angle ϕ for different values of therotational speed, i.e. n ≈ 500 rpm in (I), n ≈ 1000 rpm in(II) and n ≈ 1500 rpm in (III). From Fig. 13(a) it is apparentthat the motor current ia and its reference trajectory ida arein very good accordance. This is further confirmed by thecurrent error ∆ia in Fig. 13(b). The required control inputs,converted into duty cycles, are depicted in Fig. 13(c). It canbe seen that a large part of the control input (i.e. basically thefundamental wave component) is generated by the flatness-based feedforward controller (17). One can further observe thatthe control input demand increases with increasing rotor speed.In (III) at ϕ = 216, however, the desired torque changesits sign, cf. Fig. 12(a) at t ≈ 250 ms. As a consequence, areduction of the control inputs occurs. Please also note thatthe maximum rotor speed in Fig. 12(d) has been carefullychosen such that the maximum ratings of the VSI (i.e. dutycycles within the boundary 0 and 1) are met. In conclusion,very good control performance and practically feasible controlinputs have been obtained with the proposed flatness-basedtorque control strategy.

The results of the vector control strategy, i.e. the phasecurrent ia,dq and its reference trajectory ida,dq , are depicted inFig. 14(a). As can be seen, the vector control strategy fails totrack the desired reference trajectory in case of high desiredtorques due to the high magnetic saturation. There are two




main reasons for this undesired and unstable behavior:I) The magnetic saturation in the motor yields a load-

dependent phase shift of the flux linkages with respectto the currents, and the influence of harmonics increases.Thus, the prerequisites of the Blondel-Park transforma-tion are no longer fulfilled. Even worse, the frequency ofthe harmonics is increased by the transformation, and theadditional phase shift results in a coupling of the d-axisand q-axis1.

II) Magnetic saturation yields a change in the effectiveinductances Lm and Ls, see Fig. 9(b). Typically, theparameters of vector control concepts are chosen suchthat a good control performance is obtained for nominaltorque (this has also been done in the experiments ofthis section). This, however, results in too high controllergains in the saturated case, where Ls and Lm decrease,such that the closed loop system can become unstable.Of course, it would be possible to choose the controllergains such that stable operation is also guaranteed inthe saturated case. This, in turn, would lead to anunsatisfactory control performance in the unsaturatedcase.

It is worth mentioning that, although the identified nonlinearcurrent-to-torque characteristics given in Fig. 3 has alreadybeen considered in the classical vector control scheme used tovalidate the proposed flatness-based torque control strategy,the accuracy could be slightly increased if the nonlinearrelationship for the inductances given in Fig. 9 is used in thefeedback path of the vector controller to preserve a closed-loopdynamics, which is almost independent from the operatingpoint. Such methods are frequently used in literature. However,the main reason for the fluctuations in the torque and the dq-axis currents is most likely related to the current-dependentphase shift in Fig. 8, which is incorrectly represented in adq0-model (and extensions of it) in the presence of magneticsaturation of the motor. Moreover, the vector controller for-mulated in the rotor-fixed reference frame has to deal withdisturbances of increased frequency in comparison to a currentcontroller defined in the stator-fixed reference frame.

In conclusion, the experimental results show a very goodcontrol performance of the proposed flatness-based torquecontrol concept in the whole operating range, while classicalvector control strategies have significant drawbacks. The MECmodel, which is able to systematically capture the magneticsaturation and the non-sinusoidal behavior, serves as a basis forthe design of the flatness-based torque controller. These resultsalso show that control strategies based on dq0-models orextensions of it (either to approximately account for harmonicsor magnetic saturation) are not the best choice for applicationswhere the motor is operated in regions with significant mag-netic saturation.

VI. CONCLUSION

In this paper, a flatness-based torque control strategy for thewhole operating range of saturated surface-mounted permanent

1Note that one reason why this problem does not occur in the proposedflatness-based torque control concept is that the (physical) currents ia and ibare directly controlled.

magnet synchronous machines was presented. The proposedcontrol strategy exhibits a two-degrees-of-freedom controlstructure with a flatness-based feedforward controller and atime-variant feedback controller for the trajectory error system.Different to existing works, magnetic saturation is consideredin a thorough and physically consistent way. The proposedcontroller is formulated in the stator-fixed reference frame.Hence there is no need to utilize a coordinate transformationwhich depends on the rotor angle. Based on a calibratedmagnetic equivalent circuit model of the PMSM, which hasbeen proposed in a previous publication [35], a simplifiedmodel suitable for flatness-based control design was derived bycurrent shape optimization and a comprehensive analysis of thecorresponding flux linkages. The controller gains of the time-variant error controller, which is composed of a proportionaland integral term, systematically accounts for the mutualcoupling of the phase windings. As a benchmark experiment,the PMSM was accelerated with about four times the ratedtorque. The resulting currents generate a sufficiently smoothtorque and are in very good accordance with their referencetrajectories, even at higher speeds. It has been shown that theproposed control scheme outperforms the industrial state-of-the-art vector control implementation, which was extended bya nonlinear current-to-torque characteristic.

To increase the speed range of the PMSM beyond ratedvalues, future work is concerned with the extension of theproposed control strategy to systematically account for fieldweakening.

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial supportprovided by Bernecker & Rainer Industrie Elektronik GmbH.

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[35] D. Faustner, W. Kemmetmuller, and A. Kugi, “Magnetic equivalentcircuit modeling of a saturated surface-mounted permanent magnetsynchronous machine,” in Proc. of the 8th Vienna Int. Conferenceon Mathematical Modelling (MATHMOD), Vienna, Austria, Feb2015. [Online]. Available: http://dx.doi.org/10.3182/20150218-3-AU-30250//978-3-902823-71-70040

David Faustner (StM’14) received the Dipl.-Ing.degree in electrical engineering from TechnischeUniversitat Wien (TU Wien), Vienna, Austria, inOctober 2011. Since November 2011 he works asa research assistant at the Automation and ControlInstitute at TU Wien.His research interests include the physics basedmodeling and the nonlinear control of electricalmachines.

Wolfgang Kemmetmuller (M’04) received theDipl.-Ing. degree in mechatronics from the Jo-hannes Kepler University Linz, Austria, in 2002 andhis Ph.D. (Dr.-Ing.) degree in control engineeringfrom Saarland University, Saarbruecken, Germany,in 2007. From 2002 to 2007 he worked as a researchassistant at the Chair of System Theory and Au-tomatic Control at Saarland University. From 2007to 2012 he has been a senior researcher at the Au-tomation and Control Institute at TU Wien, Vienna,Austria and since 2013 he is Assistant Professor.

His research interests include the physics based modeling and the nonlinearcontrol of mechatronic systems with a special focus on electrohydraulicand electromechanical systems where he is involved in several industrialresearch projects. Dr. Kemmetmuller is associate editor of the IFAC journalMechatronics.

Andreas Kugi (M’94) received the Dipl.-Ing. de-gree in electrical engineering from Graz Universityof Technology, Austria, and the Ph.D. (Dr. techn.)degree in control engineering from Johannes KeplerUniversity (JKU), Linz, Austria, in 1992 and 1995,respectively. From 1995 to 2000 he worked as anassistant professor and from 2000 to 2002 as anassociate professor at JKU. He received his ”Ha-bilitation” degree in the field of automatic controland control theory from JKU in 2000. In 2002, hewas appointed full professor at Saarland University,

Saarbrucken, Germany, where he held the Chair of System Theory andAutomatic Control until May 2007. Since June 2007 he is a full professor forcomplex dynamical systems and head of the Automation and Control Instituteat TU Wien, Vienna, Austria.

His research interests include the physics-based modeling and control of(nonlinear) mechatronic systems, differential geometric and algebraic methodsfor nonlinear control, and the controller design for infinite-dimensionalsystems. He is involved in several industrial research projects in the field ofautomotive applications, electric and pneumatic servo-drives, smart structuresand rolling mill applications. Prof. Kugi is Editor-in Chief of the IFAC journalControl Engineering Practice and full member of the Austrian Academy ofSciences.



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